Fault location using measurements from two ends of a line

ABSTRACT

The present invention relates to a method to locate a fault in a section of a transmission line using measurements of current, voltage and angles between the phases at a first (A) and a second (B) end of said section. The invention is characterised by the steps of, after the occurrence of a fault along the section, calculating a distance (d A , d B ) to a fault dependent on a fault current measured at one of said first and second ends and phase voltages measured at both of said first and second ends (A, B), where the distance to fault is calculated from the end (A or B) where the fault current is measured. The invention is particularly suitable when a current transformer at either of the first or second ends (A, B) is saturated. If so, then, a distance (d) to a fault is calculated dependent on a fault current measured at the non-affected end and phase voltages measured at both the affected end and the non-affected end.

TECHNICAL FIELD

The present invention is concerned with a method and device for locatinga fault on a section of a power transmission line. The method utilizesmeasurements of current and voltage made at relays installed atterminals at both ends of the section of the power line.

BACKGROUND ART

Several methods and approaches for fault location in high voltage powersystems have been developed and employed. One approach has been to usevoltage/current transducers located at terminals, between which thepower lines to be monitored run. Inductive current transformers are usedto provide a measurement of instantaneous current in a transmissionline.

However, inductive current transformers (CTs) may become saturated undertransmission line faults with high fault current, which often takesplace during faults close to CTs installation point. The saturation ispossible especially for faults in which there is a slowly decaying dccomponent in the fault current.

Saturation of CTs influences operation of protective relays as well asthe accuracy of fault location for inspection-repair purposes. Note thataccuracy of both one-end and two-end fault locators can be adverselyaffected by the saturated CTs. U.S. Pat. No. 4,559,491 which is titledMethod and device for locating a fault point on a three-phase powertransmission line, discloses a method and device wherein a single endfault locator uses measurements of voltages and currents from aparticular side [1], and, if at the side where the fault locator isinstalled the CTs are saturated, the achieved accuracy of fault locationcould be unsatisfactory.

Better conditions may be obtained for the above single end faultlocation for the case when the CTs are saturated at the terminal whichis opposite to the fault locator installation point. In such cases theinput post fault currents of the fault locator are not contaminated dueto saturation of CTs. However, greater accuracy of fault location insuch cases can be achieved if impedance of a source from the remote side(where CTs are saturated) is known. The remote source impedance cannotbe determined with one-end measurements and therefore in someapplications the one-end fault locator may be augmented by inputting avalue for the remote end impedance. This value may be measured by theother remote device and sent via a communication channel. Note that inthis case the measured remote source impedance can differ greatly fromthe actual value due to saturation of CTs. Using an inaccuratelymeasured source impedance could deteriorate substantially the faultlocation accuracy.

Similarly, accuracy of fault location with two-end methods, as forexample with the representative methods disclosed in U.S. Pat. No.5,455,776 which is titled Automatic fault location system, and in U.S.Pat. No. 6,256,592B1 [2-3] which is titled Multi-ended fault locationsystem, are also affected by saturation of CTs. The method of U.S. Pat.No. 5,455,776 [2] uses symmetrical components of voltages and currentsfrom both sides of a line. In case of the method disclosed in U.S. Pat.No. 6,256,592B1 [3] the amplitude of the remote current and theamplitude of the remote source impedance, both determined for thenegative sequence, are utilized for calculating a distance to fault. Thedistortion of the currents, resulting from any saturated CTs, affectsthe accuracy of both the above two-end fault location techniques [2-3].No countermeasures against the possible effects of saturation aredisclosed in the cited methods [2-3].

SUMMARY OF THE INVENTION

The aim of the present invention is to remedy the above mentionedproblems.

This is obtained by a method according to claim 1 and a device forcarrying out the method according to claim 12. Specific features of thepresent invention are characterised by the appending claims.

New fault location algorithms have been derived according to anembodiment of the present invention. The algorithm utilizes post faultmeasurements of voltages from both the line section ends and post faultcurrent from one end only of the line section. The synchronized orunsynchronized measurements can be used. In case of unsynchronizedmeasurements there is a need for synchronizing the measurements in orderto provide a common time base for all the measurements. This can beobtained by introducing the term e^(jδ), where δ is the synchronizationangle, calculated from pre-fault measurements or post-fault measurementsfrom the sound phases.

The present invention presents an entirely different solution to theproblem of adverse influence of the saturation of CTs in relation tofault location. The new two-end fault location technique hereindescribed is immune to problems caused by saturation of CTs. To achievethis aim the redundancy of information contained in the voltages andcurrents, measured at both the terminals of a transmission line, hasbeen explored. It is important that the exploring of the redundancy isdone in such a way that the post-fault currents from a saturated CT arecompletely ignored and thus not used in calculations for determining adistance to fault. Contrary to that, the currents from the opposite sideof a line, the non-affected end where CTs do not saturate, are used tocalculate a distance to fault. Such an approach is possible under theabove-mentioned assumption that CTs may be saturated at one end of aline only. A known means may be used for determining whether a currenttransformer either a first or and second end (A, B) of the line issaturated, as described in more detail below. In contrast tomeasurements of current, measurements of voltages acquired post-fault atboth the terminals of a transmission line are utilized in the locationprocedure proposed by the invention.

The main advantage of the fault location algorithm according to thepresent invention is that adverse influence of CT saturation on faultlocation accuracy is avoided by using post fault currents from onenon-affected end as the input signals, i.e. from the end where thesaturation is not detected, while utilizing post fault voltages fromline terminals at both ends. Among the other advantages of the inventionare that impedances of equivalent systems behind both the line ends arenot required to be known; and that the form of the algorithm is compactbecause a first order formula has been obtained.

The information in the form of a result for the distance to a fault(d_(A) or d_(A-comp), d_(B) or d_(B-comp) or d) generated by the faultlocation method, device or system may also be embodied as a data signalfor communication via a network. The data signal may also be used toprovide a basis for a control action. The distance to a fault may besent as a signal for a control action such as: automatic notification tooperational network centres of fault and it's location or toautomatically start calculations to determine journey time to location,which repair crew shall be dispatched to site, possible time taken toexecute a repair, calculate which vehicles or crew member may be needed,how many shifts work per crew will be required and the like actions.

In another aspect of the invention a computer program product isprovided on a computer readable media which carries out steps of themethod of the invention.

In an advantageous embodiment greater accuracy for measurements on longsections or long lines may be achieved by incorporating compensation forshunt capacitances of a line. The distributed long line model isutilized for that purpose.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of the method and device of the presentinvention may be gained by reference to the following detaileddescription when taken in conjunction with the accompanying drawingswherein:

FIG. 1 shows a flowchart for a method for locating a fault according toan embodiment of the invention.

FIG. 2 shows a schematic diagram for a method for locating a fault in asection of transmission line A, B during which fault a currenttransformer is saturated at B.

FIG. 3 shows a schematic diagram for a method for locating a fault as inFIG. 2 but in which a current transformer is saturated at A.

FIG. 4 shows a flowchart for a method for locating a fault during whicha current transformer is saturated at B according to an embodiment ofthe invention.

FIG. 5 shows a flowchart for the method of FIG. 4 but in which a currenttransformer is saturated at A.

FIG. 6 shows a schematic diagram for an equivalent circuit for a sectionof a transmission line for a positive sequence component of a totalfault current, during which fault a current transformer is saturated atB.

FIG. 7 shows a schematic diagram as in FIG. 6 but for the equivalentcircuit for a negative sequence component of a total fault current.

FIG. 8 shows a schematic diagram for an equivalent positive sequencecircuit diagram for section A-B including taking into account the shuntcapacitances effect for a first iteration according to anotherembodiment of the invention.

FIG. 9 shows a schematic diagram similar to FIG. 8 for a negativesequence circuit, with taking into account the shunt capacitances effectfor a first iteration.

FIG. 10 shows a schematic diagram similar to FIG. 8, 9 for a zerosequence circuit, taking into account the shunt capacitances effect fora first iteration.

FIG. 11 shows a schematic diagram for an equivalent circuit for asection of a transmission line for a positive sequence component of atotal fault current, during which fault a current transformer issaturated at A.

FIG. 12 shows a schematic diagram as in FIG. 11 but for the equivalentcircuit for a negative sequence component of a total fault current.

FIG. 13 shows a lumped π-model of a line for the pre-fault positivesequence of the current for the purpose of calculating a term related tothe synchronisation angle (δ), according to an embodiment of theinvention.

FIG. 14 shows a block diagram for a calculation of the positive sequencephasors dependent on measurements from each end of the section A and Brespectively.

FIGS. 15, 16 a, 16 b, 17, 18 a, 18 b show schematic diagrams of possiblefault-types with respect to derivation of coefficients for Table 1A,Table 2 in Appendix 1. FIG. 15 show faults from a-g, and FIGS. 16 a, 16b faults between phases a-b. FIG. 17 shows an a-b-g fault. FIGS. 18 aand 18 b show symmetrical faults a-b-c and a-b-c-g respectively.

FIG. 19 shows details of a fault locator device according to anembodiment of the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 shows a method in the form of a flow chart according to anembodiment of the invention. The flow chart shows means 1 for receivinginput signals, a decision step 2 to determine if a CT is at one end A issaturated, and a second decision step 3 to determine whether a CT at Bis or is not saturated (when step 2=Yes). A result stage 4 is shown forwhen Fault Location FL_A shall be used, result stage 5 when FaultLocation FL_B shall be used, and result stage 6 when any Fault locationalgorithm including FL_A or FL_B may be used.

A determination of whether a CT is saturated or not may be carried outusing a method disclosed in EP 506 035 B1 entitled Method and device fordetecting saturation in current transformers, or by any other knownmethod. The method disclosed in EP 506 035 B1 is dependent oncontinuously determining an absolute value of both the current and of aderivative of the current. Three criteria calculated from the measuredand derived values are disclosed which, when satisfied simultaneously,determine that a current transformer is saturated.

FIG. 2 shows a section of a transmission line with points A and B.Included in the figure are a CTs 10, 12 and voltage transformers 11, 13.A means of communication 14 between the two ends A and B is shown. Afault F is shown at a distance d_(A) from end A. Pre fault currents I_(A) _(—) _(pre) are shown at the A end and the B end I _(B) _(—) _(pre)Post-fault current I _(A) and post-fault voltage V _(A) is shown at theA end, and a post-fault voltage V _(B) only is shown at the B end. Theimpedance of the section A to B is shown composed in part by impedance15, equal to d_(A) Z _(L) for the part from A to the fault F; and byimpedance 16, equal to (1−d_(A))Z _(L) for the part from end B to faultF. A fault locator procedure 17 is shown.

FIG. 3 shows the same arrangement essentially as in FIG. 2 but with oneor more CTs 10′ saturated at the A end, with pre fault currents andpost-fault currents and post-fault voltages marked accordingly.

In FIG. 2 the CT 12 at the B end is saturated. The pre-fault current atB, I _(B) _(—) _(pre) is disregarded, as indicated by the dotted linefrom the saturated CT 12 to the communication link 14.

FIG. 1 presents the concept of fault location when assuming saturationof CTs at one end of a section of a transmission line. Fault location isperformed on the base of three phase voltages and currents from asubstation at A (V _(A), I _(A)) and from a substation at B (V _(B), I_(B)). The method of fault location shown in FIG. 1, embodied as thefault locator procedure 17 shown in FIGS. 2, 3 may be carried out by afault locator device 20 described below with reference to FIG. 19.

The term “CTs are saturated” is understood to mean: “at least one out ofthree CTs, installed at the particular end of a section of atransmission line, is saturated”. Simultaneous magnetic saturation ofCTs at both the terminals of a transmission line is assumed not to occurin a real transmission network.

The following cases with respect to saturation of one or more CTs haveto be taken into account:

1. A CT is saturated at the side B—the fault location procedure FL_A,operating according to the proposed new method, has to be used, see FIG.2.

2. A CT is saturated at the side A—the fault location procedure FL_B,operating according to the proposed new method, has to be used, see FIG.3.

3. CTs are not saturated at both the sides of a transmission line—any ofthe fault location procedures FL_A or FL_B (operating according to theproposed new method) could be used. Referring to step 6 of FIG. 1.However, it is also possible to use any other one-end or two-end faultlocation algorithm in this case of no saturation. A two-end faultlocation method is disclosed in application SE 0004626-8 entitled Methodand device of fault location. The method includes calculating a distance(d) to a fault using the positive sequence phasors, or positive sequencequantities of post fault current and voltage measurements made at bothends of a line. The information about what type of fault has occurred,see fault types in FIGS. 15, 16 a, 16 b, 17, 18 a, 18 b, may be used todetermine which algorithm or part algorithm is used to calculate thedistance to the fault. In the case when the fault is not a 3-phasebalanced fault the distance (d) to a fault for example may also becalculated using the negative sequence quantities of post fault currentand voltage measurements made at both ends of a line. In the case whenthe fault is a 3-phase balanced fault the distance (d) to the fault may,for example, be calculated using the incremental positive sequencequantities of current and voltage measurements made at both ends of aline. The particular incremental positive sequence component isunderstood as the difference between the post fault and the pre faultvalues. These methods may be used when it is determined that CTs are notsaturated at both the sides of a transmission line to calculate distanceto a fault as per step 6, FIG. 1.

A distance to fault, obtained in a particular case, is denoted here as:

d_(A) [pu]—for the procedure FL_A used when CTs at the end B aresaturated;

d_(B) [pu]—for the procedure FL_B used when CTs at the end A aresaturated;

d [pu]—for the case of no saturation of CTs at both the ends.

As shown in FIG. 1 there are two procedures FL_A and FL_B which may becarried out according to the present invention. These procedures aredesignated to locate faults under detection of saturation of CTs at theend B and the end A, respectively. Detailed principles of fault locationwith using FL_A and FL_B procedures are shown in FIG. 2 and 3,respectively. The fault locator procedure 17 is assumed here as carriedout at the substation A. Required signals from the remote substation (B)are sent via the communication channel 14.

It is also possible to install the fault locator at the substation B. Inthis case the communication facility for sending signals from thesubstation A has to be provided. The proposed method of fault locationitself does not depend on which the arrangement is actually applied.

The two-end fault location method provided is suitable for bothsynchronized and unsynchronized measurements. In case of providingsynchronized measurements the sampled data from both the line terminalshave naturally the common time base and thus the synchronization angleequals to zero (δ=0).

In another embodiment of the invention, where in contrast to the firstembodiment, the sampling at the line terminals is run unsynchronously.In this embodiment, the measured phasors do not have a common time base.In order to provide a such common base the synchronized angle (δ≠0) hasto be introduced. In this case, for unsynchronized measurements, theintroduced synchronized angle may be any value and thus it has to bedetermined from the available measurements. For this purpose the faultlocator procedure 17 has to be supplied with the pre-fault phasecurrents (FIG. 2, 3—shown as an input data of the fault locator that ismarked with the dashed lines), which allow calculation of thesynchronization angle.

The synchronization angle (δ) is introduced in the form of an agente^(jδ), which is multiplied by the phasors of phase voltages andcurrents acquired at a particular substation: for the procedure FL_A(FIG. 2)—the phasors from the substation A are multiplied by e^(jδ), forthe procedure FL_B (FIG. 3)—the phasors from the substation B aremultiplied by e^(jδ).

The procedure FL_A (the case of saturation of CTs at the side B—FIG. 2and FIG. 4) applies the following measurements of phasors:

for determining a distance to fault (d_(A)):

I _(A)—post-fault currents from the side A from particular phases a, b,c:I _(A) _(—) _(a), I _(A) _(—) _(b), I _(A) _(—) _(c)

V _(A)—post-fault voltages from the side A from particular phases a, b,c:V _(A) _(—) _(a), V _(A) _(—) _(b), A _(—) _(c)

V _(B)—post-fault voltages from the side B from particular phases a, b,c:V _(B) _(—) _(a), V _(B) _(—) _(b), V _(B) _(—) _(c)

for determining a synchronization angle (δ) in case of no providing thesynchronization of measurements (for synchronized measurements: δ=0):

I _(A) _(—) _(pre), pre-fault currents from the side A from particularphases a, b, c:I _(A) _(—) _(pre) _(—) _(a), I _(A) _(—) _(pre) _(—) _(b), I _(A) _(—)_(pre) _(—) _(c)

I _(B) _(—) _(pre)—pre-fault currents from the side B from particularphases a, b, c:I _(B) _(—) _(pre) _(—) _(a), I _(B) _(—) _(pre) _(—) _(b), I _(B) _(—)_(pre) _(—) _(c)

It follows by correspondence then that the procedure FL_B (the case ofsaturation of CTs at the side A—FIG. 3 and FIG. 5) applies the followingmeasurements of phasors:

for determining a distance to fault (d_(B)):

I _(B)—post-fault currents from the side B from particular phases a, b,c:I _(B) _(—) _(a), I _(B) _(—) _(b), I _(B) _(—) _(c)

V _(A)—post-fault voltages from the side A from particular phases a, b,c:V _(A) _(—) _(a), V _(A) _(—) _(b), V _(A) _(—) _(c)

V _(B)—post-fault voltages from the side B from particular phases a, b,c:V _(B) _(—) _(a), V _(B) _(—) _(b), V _(B) _(—) _(c)

1for determining a synchronization angle (δ) in case of no provision forthe synchronization of measurements (for synchronized measurements: δ0):

I _(A) _(—) _(pre)—pre-fault currents from the side A from particularphases a, b, c:I _(A) _(—) _(pre) _(—) _(a), I _(A) _(—) _(pre) _(—) _(b), I _(A) _(—)_(pre) _(—) _(c)

I _(B) _(—) _(pre)—pre-fault currents from the side B from particularphases a, b, c:I _(B) _(—) _(pre) _(—) _(a), I _(B) _(—) _(pre) _(—) _(b), I _(B) _(—)_(pre) _(—) _(c)

Referring to FIG. 4. FIG. 4 shows in box 31 the input data, includingcurrent, voltage, line impedance measurements and a fault-type input. Adecision step 32 determines if the measurements are synchronised. Adecision of NO, (δ≠0), leads to the synchronisation angle (δ) beingcalculated in box 33. A decision of YES leads to box 34 for adaptivefiltering of phase quantities, calculation of symmetrical components ofvoltages and currents. Box 35 calculates a value for location of thefault without taking into account shunt capacitance effects. The value,d_(A), is available as a result 7 for distance to a fault.

In a further embodiment of the invention, Box 37 receives capacitancevalues and line length 1 as input and calculates a distance from A to afault d_(A) _(—) _(comp) with compensation for shunt capacitance. FIG. 5shows a corresponding diagram for the case when CTs are saturated at Aand a distance from B to a fault d_(B), is available at 7′ and adistance to a fault with compensation for shunt capacitance d_(B) _(—)_(comp) is available at 9 c.

Besides the above listed input signals described with reference to FIGS.1-3, both the procedures (FL_A and FL_B) require the followingparameters shown if FIG. 4, 5: fault-type—this information can beprovided from a protection system or a dedicated classificationprocedure can be incorporated,

Z _(L1)—impedance of a whole line for the positive (negative) sequence,

Z _(L0)—impedance of a whole line for the zero sequence

l—line length (km)

C_(L1)—shunt capacitance of a whole line for the positive (negative)sequence

C_(L0)—shunt capacitance of a whole line for the positive zero sequence.

Two of the last three parameters, (l, and C_(L1) or C_(L0)) may berequired for introducing the compensation for shunt capacitances of atransmission line (according to distributed long line model of Andersson[4]) in the further embodiment of the invention. Under an assumptionthat the positive sequence capacitance is identical to the negativesequence under pre-fault conditions a value either for C_(L1) or C_(L0)may be used. A distance to fault after the compensating for shuntcapacitances is denoted as: d_(A) _(—) _(comp) (FIG. 4) and d_(B) _(—)_(comp) (FIG. 5), respectively.

In order to derive this location procedure (see FIG. 2, 4) the faultcurrent distribution factors have to be considered. As it will be shownin details below, it is sufficient to consider these factors only forthe positive and negative sequence. FIG. 6 presents the equivalentcircuit diagram of a transmission line for the positive sequence, andFIG. 7 presents the equivalent circuit diagram for the negativesequence. At this stage of the derivation the shunt parameters of a lineare neglected. The terminals of a line are denoted by A and B. The faultpoint is marked by F.

Positive sequence component of a total fault current (FIG. 6) is thefollowing sum:I _(F1) =I _(A1) _(e) ^(jδ) +I _(B1)   (1)

Thus, positive sequence current I _(B1) can be expressed:I _(B1) =I _(F1) −I _(A1) _(e) ^(jδ)  (2)

Considering the voltage drop between the busbars A and B, with takinginto account (2), we obtain:V _(A1) ^(jδ) −d _(A) Z _(L1) I _(A1) _(e) ^(jδ) =V _(B1) —(1−d _(A)) Z_(L1)(I _(F1) −I _(A1) ^(jδ))   (3)

Fault current from (3) is determined as: $\begin{matrix}{{{\underset{\_}{I}}_{F\quad 1} = \frac{{\underset{\_}{M}}_{1\quad A}}{1 - d_{A}}}{{where}\text{:}}{{\underset{\_}{M}}_{1\quad A} = {\frac{{{- {\underset{\_}{V}}_{A\quad 1}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{B\quad 1}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{A\quad 1}{\mathbb{e}}^{j\delta}}}}} & (4)\end{matrix}$

(δ)=synchronization angle, introduced for providing the common time basefor measurements acquired at different ends of a transmission line.

Thus, positive sequence component of the total fault current isexpressed by measurements from side A (V _(A1), I _(A1)) and from side B(V _(B1)-only). Measurements at the side B are taken here as the basisand thus measurements from the side A are taken into account with thesynchronization angle (δ). In case of the synchronized measurements wehave: δ=0. For the unsynchronized measurements this angle is unknown(δ≠0) and has to be determined by utilizing relations, which are validfor pre-fault currents or for post-fault currents but from the healthyphases. In further derivation this angle is treated as of the knownvalue.

Analogously we have for the negative sequence (FIG. 7): $\begin{matrix}{{{\underset{\_}{I}}_{F\quad 2} = \frac{{\underset{\_}{M}}_{2\quad A}}{1 - d_{A}}}{{where}\text{:}}{{\underset{\_}{M}}_{2A} = {\frac{{{- {\underset{\_}{V}}_{A\quad 2}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{B\quad 2}}{{\underset{\_}{Z}}_{L\quad 2}} + {{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}}}}} & (5)\end{matrix}$

Z _(L2)=Z _(L1)—impedance of a line for the negative is the same as forthe positive sequence.

Analogously we can determine the zero sequence component of the totalfault current (I _(F0)). However, this quantity will involve theimpedance of a line for the zero sequence (Z _(L0)) Since this impedance(Z _(L0)) is considered as uncertain parameter, thus, I _(F0) isrecommended as not to be used when representing the voltage drop acrossa fault resistance (this concept is taken from the fault locatorpresented in [1]).

The generalized fault loop model is utilized for deriving the consideredfault location procedure. This is a single formula with the coefficientsdependent on a fault type, covering different fault types. In words thisformula may be written: [fault loop voltage] minus [voltage drop acrossthe faulted segment of a line] minus [voltage drop across the faultresistance] equals zero. The actual formula is: $\begin{matrix}{\lbrack {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{A\quad 0}{\mathbb{e}}^{j\delta}}} \rbrack - {\quad{\lbrack {d_{A}{{\underset{\_}{Z}}_{L\quad 1}( {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}{\mathbb{e}}^{j\delta}}} )}} \rbrack + \ldots - {\quad{\lbrack {R_{F}( {{{\underset{\_}{a}}_{F\quad 1}\frac{{\underset{\_}{M}}_{1\quad A}}{1 - d_{A}}} + {{\underset{\_}{a}}_{F\quad 2}\frac{{\underset{\_}{M}}_{2\quad A}}{1 - d_{A}}} + {{\underset{\_}{a}}_{F\quad 0}\frac{{\underset{\_}{M}}_{0\quad A}}{1 - d_{A}}}} )} \rbrack = 0}}}}} & (6)\end{matrix}$

where: α ₁, α ₂, α ₀, α _(F1), α _(F2), ′ _(F0)—the coefficientsdependent on a fault type, gathered in TABLEs 1, 2. The derivation ofthe listed in the following Tables 1, 2 coefficients is presented inAPPENDIX 1. TABLE 1 Coefficients for determining fault loop signalsdefined. Fault type a₁ a₂ a₀ a-g 1 1 1 b-g a² a 1 c-g a a² 1 a-b, a-b-g1 − a² 1 − a 0 a-b-c, a-b-c-g b-c, b-c-g a² − a a − a² 0 c-a, c-a-g a −1 a² − 1 0 a = exp(j2π/3)

TABLE 2 Alternative sets of the weighting coefficients from (5) fordetermining a voltage drop across the fault path resistance. Fault Set ISet II Set III type a_(F1) a_(F2) a_(F0) a_(F1) a_(F2) a_(F0) a_(F1)a_(F2) a_(F0) a-g 0 3 0 3 0 0 1.5 1.5 0 b-g 0 3a 0 3a² 0 0 1.5a² 1.5a 0c-g 0 3a² 0 3a 0 0 1.5a 1.5a² 0 a-b 0 1 − a 0 1 − a² 0 0 0.5(1 − a²)0.5(1 − a) 0 b-c 0 a − a² 0 a² − a 0 0 0.5(a² − a) 0.5(a − a²) 0 c-a 0a² − 1 0 a − 1 0 0 0.5(a − 1) 0.5(a² − 1) 0 a-b-g 1 − a² 1 − a 0 1 − a²1 − a 0 1 − a² 1 − a 0 b-c-g a² − a a − a² 0 a² − a a − a² 0 a² − a a −a² 0 c-a-g a − 1 a² − 1 0 a − 1 a² − 1 0 a − 1 a² − 1 0 a-b-c-g 1 − a² 00 1 − a² 0 0 1 − a² 0 0 (a-b-c)

Voltage drop across the fault path (as shown in the third term ofequation (6)) is expressed using sequence components of the total faultcurrent. The weighting coefficients (α _(F0), α _(F1), α _(F2)) canaccordingly be determined by taking the boundary conditions for aparticular fault type. However, there is some freedom for that. Thus, itis proposed firstly to utilize this freedom for avoiding zero sequencequantities. This proposal has been taken since the zero sequenceimpedance of a line is considered as an unreliable parameter. Avoidingthe zero sequence impedance of a line can be accomplished here bysetting α _(F0)=0 as shown in TABLE 2. Secondly, the freedom inestablishing the weighting coefficients can be utilized for determiningthe preference for using particular quantities. Thus, the voltage dropacross the fault path is expressed further with using positive andnegative sequence quantities only (TABLE 2).

There are two unknowns: d_(A), R_(F) in equation (6). Note that thesynchronization angle (δ), as mentioned above, is known as:

δ=0—for the synchronized measurements or

δ≠0—for the unsynchronized measurements;

where the synchronization angle is determined from the measurements(using pre-fault currents or post-fault currents but from the healthyphases).

Taking into account that in equation (6) we have adjusted α _(F0)=0 andlet us also write (6) in more compact form for further derivations:$\begin{matrix}{{{{\underset{\_}{A}}_{v} - {d_{A}{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} - {\frac{R_{F}}{1 - d_{A}}\lbrack {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} \rbrack}} = 0}{{where}\text{:}}{{\underset{\_}{A}}_{v} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{A\quad 0}{\mathbb{e}}^{j\delta}}}}{{\underset{\_}{A}}_{i} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}{\mathbb{e}}^{j\delta}}}}} & (7)\end{matrix}$

Separating equation (7) for real and imaginary parts we obtain:$\begin{matrix}{{{{real}( {\underset{\_}{A}}_{v} )} - {d_{A}{{real}( {{\underset{\_}{Z}}_{1\quad L}{\underset{\_}{A}}_{i}} )}} - {\frac{R_{F}}{1 - d_{A}}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}} = 0} & ( {8a} )\end{matrix}$ $\begin{matrix}{{{{imag}( {\underset{\_}{A}}_{v} )} - {d_{A}{{imag}( {{\underset{\_}{Z}}_{1\quad L}{\underset{\_}{A}}_{i}} )}} - {\frac{R_{F}}{1 - d_{A}}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}} = 0} & ( {8b} )\end{matrix}$

Note, that in the above equations (8a), (8b) it was considered that:$\frac{R_{F}}{1 - d_{A}} -$is a real number

Multiplying (8a) by: imag(α _(F1) M _(1A)+α _(F2) M _(2A)) and (8b) by:real(α _(F1) M _(1A)+α _(F2) M _(2A)) we obtain: $\begin{matrix}{{{{{real}( {\underset{\_}{A}}_{v} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} - {d_{A}{{real}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2A}}} )}} + {\ldots\quad\ldots} - {\frac{R_{F}}{1 - d_{A}}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F2}{\underset{\_}{M}}_{2\quad A}}} )}}} = 0} & ( {9a} ) \\{{{{{imag}( {\underset{\_}{A}}_{v} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} - {d_{A}{{imag}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} + {\ldots\quad\ldots} - {\frac{R_{F}}{1 - d_{A}}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}} = 0} & ( {9b} )\end{matrix}$

Subtracting (9b) from (9a) we cancel fault resistance R_(F) and obtainthe solution for a distance to fault in the following form:$\begin{matrix}{{d_{A} = \frac{\begin{matrix}{{{{real}( {\underset{\_}{A}}_{v} )}{{imag}( {{{\underset{\_}{a}}_{F1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {\underset{\_}{A}}_{v} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}\end{matrix}}{\begin{matrix}{{{{real}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}\end{matrix}}}{{where}\text{:}}{{\underset{\_}{A}}_{v} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{A\quad 0}{\mathbb{e}}^{j\delta}}}}{{\underset{\_}{A}}_{i} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}{\mathbb{e}}^{j\delta}}}}{{\underset{\_}{M}}_{1\quad A} = {\frac{{{- {\underset{\_}{V}}_{A\quad 1}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{B\quad 1}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{A\quad 1}{\mathbb{e}}^{j\delta}}}}{{\underset{\_}{M}}_{2\quad A} = {\frac{{{- {\underset{\_}{V}}_{A\quad 2}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{B\quad 2}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}}}}} & (10)\end{matrix}$

α ₁, α ₂, α ₀, α _(F1), α _(F2)—coefficients dependent on fault-type(TABLE 1, 2)

(δ)δsynchronization angle.

Distance to a fault (d_(A)) according to (10) is determined under thecondition of neglecting shunt capacitances of a transmission line. Incase of short lines, say up to 150 km, it is sufficient for achievinghigh accuracy of fault location.

In a further embodiment of the invention and preferably for longerlines, the shunt capacitances effect may be compensated for. Otherwise,with lines of up to say 150 km and longer the location accuracy can beconsiderably deteriorated.

Compensation for a shunt capacitance effect of a line can beaccomplished by taking into account the lumped π—model (lumped pi model)or the distributed long transmission line model. The distributed longline model, which provides greater accuracy of fault location, has beenapplied.

Fault location procedure with compensation for shunt capacitances of atransmission line requires the following additional input data, shown inFIG. 4:

C_(L1)—shunt capacitance of a whole line for the positive and thenegative sequences (parameters of a line for the positive and thenegative sequences are identical and thus: C_(L2)=C_(L1))

C_(L0) shunt capacitance of a whole line for the zero sequence,

l—total line length (km).

The compensation of shunt capacitances is introduced while determiningthe voltage drop across the faulted line segment (in this examplebetween points A and F)—the second term in the generalized fault loopmodel (6). This requires compensating the components of the measuredcurrents for particular sequences. Thus, the original measured currents:I _(A1), I _(A2), I _(A0) have to be replaced by the currents after theintroduced compensation: I _(A1) _(—) _(comp), I _(A2) _(—) _(comp), I_(A0) _(—) _(comp). At the same time the original fault loop voltage(the first term in the model (6)) is taken for a distance to faultcalculation. As concerns determining the voltage drop across the faultresistance (the third term in (6)), it is assumed here, which is astandard practice, that the effect of line capacitances at the faultlocation (point F) may be neglected. This is justified because theimpedance of the capacitive branch at that location is much greater thanthe fault resistance. This means that the voltage drop across the faultresistance is determined without taking into account the shuntcapacitances.

Using the above assumptions for the compensation of line capacitancesthe formula for a distance to fault (10) is modified to the followingform: $\begin{matrix}{{d_{A\_ comp} = \frac{\begin{matrix}{{{{real}( {\underset{\_}{A}}_{v} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {\underset{\_}{A}}_{v} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}\end{matrix}}{\begin{matrix}{{{{real}( {{\underset{\_}{Z}}_{L\quad 1}^{long}{\underset{\_}{A}}_{i\_ comp}} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {{\underset{\_}{Z}}_{L\quad 1}^{long}{\underset{\_}{A}}_{i\_ comp}} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}\end{matrix}}}{{where}\text{:}}{{{\underset{\_}{A}}_{i\_ comp} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1{\_ comp}}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2{\_ comp}}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}^{long}}{{\underset{\_}{Z}}_{L\quad 1}^{long}}{\underset{\_}{I}}_{A\quad 0{\_ comp}}{\mathbb{e}}^{j\delta}}}},}} & (11)\end{matrix}$

I _(A1) _(—) _(comp), I _(A2) _(—) _(comp), I _(A0) _(—)_(comp)—positive, negative and zero sequence currents after thecompensation,

Z _(L1) ^(long)—positive sequence impedance of a line with taking intoaccount the distributed long line model (will be defined below at thepoint where the compensation for the currents will be presented),

Z _(L0) ^(long)—as above, but for the zero sequence, the otherquantities are defined in (10).

The compensation procedure requires iterative calculations, performeduntil the convergence is achieved (i.e. until the location estimatescease to change from the previous estimates). However, the studiesconducted by the inventors revealed that results of acceptable accuracyare obtained using 2-3 iterations, thus a fixed number of iterations.The calculated distance to a fault from a particular (say, presentiteration) is utilized for determining the shunt current in the nextiteration. The determined shunt current is then deduced from themeasured current. A distance to fault, calculated without consideringthe shunt effect (10), is taken as the starting value for the firstiteration. The way of conducting the first iteration of the compensationis shown in FIGS. 8, 9, 10.

FIG. 8 is a diagram of a positive sequence circuit with taking intoaccount the shunt capacitances effect for a first iteration. FIG. 9 is anegative sequence circuit diagram and FIG. 10 is a zero sequence circuitdiagram each taking into account the shunt capacitances effect for afirst iteration.

As a result of performing the first iteration for the positive sequence(FIG. 8) the compensated current (I _(A1) _(—) _(comp) _(—) ₁; the lastindex in the subscript denotes the first iteration) is calculated. Thisis based on deducing the shunt current from the original measuredcurrent (I _(A1)):I _(A1) _(—) _(comp) _(—) ₁ =I _(A1)−0.5d _(A) lB _(L1) A _(tanh 1) V_(A1)   (12)

where:

d_(A)—distance to fault calculated under no taking into account theshunt capacitance effect (10),

l—total line length (km)${\underset{\_}{A}}_{\tanh\quad 1} = \frac{\tanh( {\sqrt{0.5{\underset{\_}{Z}}_{L\quad 1}^{\prime}{\underset{\_}{B}}_{L\quad 1}^{\prime}}d_{A}l} )}{\sqrt{0.5{\underset{\_}{Z}}_{L\quad 1}^{\prime}{\underset{\_}{B}}_{L\quad 1}^{\prime}d_{A}l}}$${\underset{\_}{B}}_{L\quad 1}^{\prime} = {\frac{{j\omega}\quad C_{L\quad 1}}{l} - {{positive}\quad{sequence}\quad{admittance}\quad({capacitive})\quad{of}\quad a\quad{line}\quad{per}\quad{km}\quad{length}\quad( {S\text{/}{km}} )}}$${\underset{\_}{Z}}_{L\quad 1}^{\prime} = {\frac{{\underset{\_}{Z}}_{L\quad 1}}{l} - {{positive}\quad{sequence}\quad{impedance}\quad{of}\quad a\quad{line}\quad{per}\quad{km}\quad{length}\quad( {\Omega\text{/}{km}} )}}$

Positive sequence impedance of a faulted line segment (between points Aand F) without taking into account the shunt capacitances effect andusing a simple R-L model, that is, a simple model excluding capacitance.For example such as a circuit equivalent to the circuit of FIG. 13without the two capacitances equals:d_(A)lZ′_(L1)   (13)

while for the distributed long line model: $\begin{matrix}{{d_{A}l{\underset{\_}{Z}}_{L\quad 1}^{\prime}{\underset{\_}{A}}_{\sinh\quad 1}}{{where}\text{:}}{{\underset{\_}{A}}_{\sinh\quad 1} = \frac{\sinh( {\sqrt{{\underset{\_}{Z}}_{L\quad 1}^{\prime}{\underset{\_}{B}}_{L\quad 1}^{\prime}}d_{A}l} )}{\sqrt{{\underset{\_}{Z}}_{L\quad 1}^{\prime}{\underset{\_}{B}}_{L\quad 1}^{\prime}}d_{A}l}}} & (14)\end{matrix}$

Thus, the positive sequence impedance of a line with taking into accountthe distributed long line model (Z _(L1) ^(long)), which has to be usedin the formula (11), equals:Z _(L1) ^(long)=A _(sinh 1) Z _(L1)   (15)

As a result of performing the first iteration for the negative sequence,FIG. 9, the compensated current (IA2 _(—) _(comp) _(—) ₁; the last indexin the subscript denotes the first iteration) is calculated. This isbased on deducing the shunt current from the original measured current(I _(A2)):I _(A2) _(—) _(comp) _(—) ₁ =I _(A2)−0.5d _(A) lB′ _(L2) A _(tanh 2) V_(A2)   (16)

where, taking into account that the line parameters for the positive andfor the negative sequences are identical (C_(L2)=C_(L1), Z _(L2)=Z_(L1)):A _(tanh 2)=A _(tanh 1)B′_(L2)=B _(L1)

As a result of performing the first iteration for the zero sequence,FIG. 10, the compensated current (IA0 _(—) _(comp) _(—) ₁; the lastindex in the subscript denotes the first iteration) is calculated. Thisis based on deducing the shunt current from the original measuredcurrent (I _(A0)): $\begin{matrix}{{{\underset{\_}{I}}_{{A0\_ comp}\_ 1} = {{\underset{\_}{I}}_{A0} - {0.5d_{A}l{\underset{\_}{B}}_{L\quad 0}^{\prime}{\underset{\_}{A}}_{\tanh\quad 0}{\underset{\_}{V}}_{A\quad 0}}}}{{where}\text{:}}{{\underset{\_}{A}}_{\tanh\quad 0} = \frac{\tanh( {\sqrt{0.5{\underset{\_}{Z}}_{L\quad 0}^{\prime}{\underset{\_}{B}}_{L\quad 0}^{\prime}}d_{A}l} )}{\sqrt{0.5{\underset{\_}{Z}}_{L\quad 0}^{\prime}{\underset{\_}{B}}_{L\quad 0}^{\prime}}d_{A}l}}{{\underset{\_}{B}}_{L\quad 0}^{\prime} = {\frac{{j\omega}\quad C_{L\quad 0}}{l} - {{zero}\quad{sequence}\quad{admittance}\quad({capacitive})\quad{of}\quad a\quad{line}\quad{per}\quad{km}\quad{length}\quad( {S\text{/}{km}} )}}}{{\underset{\_}{Z}}_{L\quad 0}^{\prime} = {\frac{{\underset{\_}{Z}}_{L\quad 0}}{l} - {{zero}\quad{sequence}\quad{impedance}\quad{of}\quad a\quad{line}\quad{per}\quad{km}\quad{length}\quad( {\Omega\text{/}{km}} )}}}} & (17)\end{matrix}$

Zero sequence impedance of a faulted line segment (between points A andF) without taking into account the shunt capacitances effect andconsidering the simple R-L model, described above such as a circuitequivalent to the circuit of FIG. 13 without the two capacitances:d_(A)lZ _(L0)   (18)

while for the distributed long line model: $\begin{matrix}{{d_{A}l{\underset{\_}{Z}}_{L\quad 0}^{\prime}{\underset{\_}{A}}_{\sinh\quad 0}}{{where}\text{:}}{{\underset{\_}{A}}_{\sinh\quad 0} = \frac{\sinh( {\sqrt{{\underset{\_}{Z}}_{L\quad 0}^{\prime}{\underset{\_}{B}}_{L\quad 0}^{\prime}}d_{A}l} )}{\sqrt{{\underset{\_}{Z}}_{L\quad 0}^{\prime}{\underset{\_}{B}}_{L\quad 0}^{\prime}}d_{A}l}}} & (19)\end{matrix}$

Thus, the zero sequence impedance of a line, taking into account thedistributed long line model (Z _(L0) ^(long)), which has to be used inthe formula (11), equals:Z _(L)) ^(long)=A _(sin 0) Z _(L0)   (20)

A method for a fault location according to the invention in the casewhere saturation occurs at the first line section end A begins with acalculation of the positive sequence component for FL_B.

Again, for deriving this location procedure (see FIGS. 3, 5) the faultcurrent distribution factors have to be considered and also it issufficient to consider these factors for the positive and negativesequence only. FIG. 8 presents the equivalent circuit diagram of atransmission line for the positive sequence, while FIG. 9 presents theequivalent circuit diagram for the negative sequence. At this stage ofthe derivation the shunt parameters of a line are also neglected.

Positive sequence component of a total fault current (FIG. 8) is thefollowing sum:I _(F1) =I _(B1) _(e) ^(jδ) +I _(A1)   (21)

Thus, positive sequence current I _(A1) can be expressed:I _(A1) =I _(F1) −I _(B1) _(e) ^(jδ)  (22)

Considering the voltage drop between the busbars at B and A, with takinginto account (22), we obtain:V _(B1) _(e) ^(jδ) −d _(B) Z _(L1) I _(B1) _(e) ^(jδ) =V _(A1)−(1−d_(B)) Z _(L1)( I _(F1) −I _(B1) _(e) ^(jδ))   (23)

Fault current from (23) is determined as: $\begin{matrix}{{{\underset{\_}{I}}_{F\quad 1} = \frac{{\underset{\_}{M}}_{1\quad B}}{1 - d_{B}}}{{where}\text{:}}{{\underset{\_}{M}}_{1\quad B} = {\frac{{{- {\underset{\_}{V}}_{B\quad 1}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{A\quad 1}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{B\quad 1}{\mathbb{e}}^{j\delta}}}}} & (24)\end{matrix}$

(δ) is synchronization angle, introduced for providing the common timebase for measurements acquired at different ends of a section of atransmission line.

Thus, positive sequence component of the total fault current isexpressed by measurements from side B (V _(B1), I _(B1)) and from side A(V _(A1)-only). Measurements at the side A are taken here as the basisand thus measurements from the side B are taken into account with thesynchronization angle (δ).

Analogously we have for the negative sequence (FIG. 9): $\begin{matrix}{{{\underset{\_}{I}}_{2} = \frac{{\underset{\_}{M}}_{2\quad B}}{1 - d_{B}}}{{where}\text{:}}{{\underset{\_}{M}}_{2\quad B} = {\frac{{{- {\underset{\_}{V}}_{B\quad 2}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{A\quad 2}}{{\underset{\_}{Z}}_{L\quad 2}} + {{\underset{\_}{I}}_{B\quad 2}{\mathbb{e}}^{j\delta}}}}} & (25)\end{matrix}$

Z _(L2)=Z _(L1)—impedance of a line for the negative is the same as forthe positive sequence.

Calculation of the zero sequence component. Analogously we can determinethe zero sequence component of the total fault current (I _(F0)).However, this quantity will involve the impedance of a line for the zerosequence (Z _(L0)). Since this impedance (Z _(L0)) is considered asuncertain parameter, thus, I ₀ is recommended as not to be used whenrepresenting the voltage drop across a fault resistance (this concept istaken from the original RANZA fault locator [1).

The generalized fault loop model is utilized for deriving the faultlocation procedure FL_B considered here: $\begin{matrix}{\lbrack {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{B\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{B\quad 2}{\mathbb{e}}^{j\delta}}} \rbrack - {\quad{{\lbrack {d_{B}{{\underset{\_}{Z}}_{L\quad 1}( {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{B\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{B2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{B\quad 0}{\mathbb{e}}^{j\delta}}} )}} \rbrack + \ldots - \lbrack {R_{F}( {{{\underset{\_}{a}}_{F\quad 1}\frac{{\underset{\_}{M}}_{1\quad B}}{1 - d_{B}}} + {{\underset{\_}{a}}_{F\quad 2}\frac{{\underset{\_}{M}}_{2\quad B}}{1 - d_{B}}} + {{\underset{\_}{a}}_{F\quad 0}\frac{{\underset{\_}{M}}_{0\quad B}}{1 - d_{B}}}} )} \rbrack} = 0}}} & (26)\end{matrix}$

where:

α ₁, α ₂, α ₀, α _(F1), α _(F2), α _(F0)—coefficients dependent on afault type (TABLE 1, 2).

Voltage drop across the fault path (as shown in the third term ofequation (16)) is expressed using sequence components of the total faultcurrent. The weighting coefficients (α _(F0), α _(F1), α _(F2)) canaccordingly be determined by taking the boundary conditions forparticular fault type. However, there is some freedom for that.Utilization of this freedom has been done in the same way as before forthe procedure FL_A. Again it is assumed: α _(F0)=0

There are two unknowns: d_(B), R_(F) in equation (26). Note that thesynchronization angle (δ), as mentioned at the beginning, is known:

δ=0—for the synchronized measurements or

δ≠0—for the unsynchronized measurements; the

synchronization angle is determined from the measurements (usingpre-fault currents or post-fault currents but from the healthy phases).

Let us take into account that in (26) we have adjusted α _(F0)=0 and letus also write (16) in more compact form for further derivations:$\begin{matrix}{{{{\underset{\_}{B}}_{v} - {d_{B}{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{B}}_{i}} - {\frac{R_{F}}{1 - d_{B}}\lbrack {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} \rbrack}} = 0}{{where}\text{:}}{{\underset{\_}{B}}_{v} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{B\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{B\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{B\quad 0}{\mathbb{e}}^{j\delta}}}}{{\underset{\_}{B}}_{i} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{B\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{B\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{B\quad 0}{\mathbb{e}}^{j\delta}}}}} & (27)\end{matrix}$

Separating (27) for real and imaginary parts we obtain: $\begin{matrix}{{{{real}( {\underset{\_}{B}}_{v} )} - {d_{B}\quad{{real}( {{\underset{\_}{Z}}_{1\quad L}{\underset{\_}{B}}_{i}} )}} - {\frac{R_{F}}{1 - d_{B}}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}}} = 0} & ( {28a} ) \\{{{{imag}( {\underset{\_}{B}}_{v} )} - {d_{B}\quad{{imag}( {{\underset{\_}{Z}}_{1\quad L}{\underset{\_}{B}}_{i}} )}} - {\frac{R_{F}}{1 - d_{B}}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}}} = 0} & ( {28b} )\end{matrix}$

Note, that in the above equations (28a), (28b) it was considered that:$\frac{R_{F}}{1 - d_{B}} -$is a real number.

Multiplying (28a) by: imag(α _(F1) M _(1B)+α _(F2) M _(2B)) and (28b)by: real (α _(F1) M _(1B)+α _(F2) M _(2B)) we obtain: $\begin{matrix}{{{{{real}( {\underset{\_}{B}}_{v} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2B}}} )}} - {d_{B}\quad{{real}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{B}}_{i}} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}} + {\ldots\ldots} - {\frac{R_{F}}{1 - d_{B}}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}}} = 0} & ( {29a} )\end{matrix}$ $\begin{matrix}{{{{{imag}( {\underset{\_}{B}}_{v} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2B}}} )}} - {d_{B}\quad{{imag}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{B}}_{i}} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}} + {\ldots\ldots} - {\frac{R_{F}}{1 - d_{B}}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}}} = 0} & ( {29b} )\end{matrix}$

Subtracting (29b) from (29a) we cancel fault resistance R_(F) and obtainthe solution for a distance to fault in the following form:$\begin{matrix}{{d_{B} = \frac{\begin{matrix}{{{{real}( {\underset{\_}{B}}_{v} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}} -} \\{{{imag}( {\underset{\_}{B}}_{v} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}}\end{matrix}}{\begin{matrix}{{{{real}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{B}}_{i}} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}} -} \\{{{imag}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{B}}_{i}} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1B}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad B}}} )}}\end{matrix}}}{{where}\text{:}}{{\underset{\_}{B}}_{v} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{B\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{B\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{B\quad 0}{\mathbb{e}}^{j\delta}}}}{{\underset{\_}{B}}_{i} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{B\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{B\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{B\quad 0}{\mathbb{e}}^{j\delta}}}}{{\underset{\_}{M}}_{1\quad B} = {\frac{{{- {\underset{\_}{V}}_{B\quad 1}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{A\quad 1}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{B\quad 1}{\mathbb{e}}^{j\delta}}}}{{\underset{\_}{M}}_{2\quad B} = {\frac{{{- {\underset{\_}{V}}_{B\quad 2}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{A\quad 2}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{B\quad 2}{\mathbb{e}}^{j\delta}}}}} & (30)\end{matrix}$

α ₁, α ₂, α ₀, α _(F1), α _(F2)—coefficients dependent on a fault-type(TABLE 1, 2)

(δ)—synchronization angle.

Compensation for shunt capacitance effects to calculate a compensatedvalue for d_(B) i.e. a value for d_(B-comp) can be accomplishedanalogously to the method presented for the procedure FL_A above.

The derived fault location procedures FL_A (10) and FL_B (20) requirehaving the measurements from the line terminals related to the commontime base. In case of the synchronized measurements it is providedautomatically. In contrast, for the unsynchronized measurements thesynchronization angle (δ≠0) plays this role. The introducedsynchronization angle for the unsynchronized measurements is unknown andthus has to be calculated from the available measurements. To be moreprecise, there is a need for calculating the term e^(jδ) and not thesynchronization angle (δ) itself. This is so since the phasors of thesymmetrical components are processed in the location procedures.

The synchronization, i.e. calculating the term e^(jδ) can be performedby utilizing pre-fault measurements.

FIG. 13 shows a lumped π model of a line for the pre-fault positivesequence including the shunt capacitances

The required term e^(jδ) can be calculated by considering the relationsvalid for the pre-fault conditions. For this purpose the shuntcapacitances of a transmission line have to be taken into account, seeFIG. 13. Note that for the shunt branches shown in FIG. 13 theadmittances (0.5B _(L1)) and thus not impedances are indicated, where: B_(L1)=jω₁C_(L1); C_(L1)-positive sequence shunt capacitance of the wholeline.

FIG. 14 is a diagram for the determination of the positive sequencephasors for the pre-fault phase currents and voltages acquired at thesubstations A and B.

In order to determine the value of e^(j) ^(δ) the computation startsfrom calculating positive sequence phasors of the pre-fault phasevoltages and currents acquired at the substations A and B, see FIG. 14.For example, taking the pre-fault currents from phases (a, b, c) at thestation A (I _(A) _(—) _(pre) _(—) _(a), I _(A) _(—) _(pre) _(—) _(b), I_(A) _(—) _(pre) _(—) _(c)) the positive sequence phasor (I _(A) _(—)_(pre) _(—) ₁) is calculated. Analogously the phase voltages from thestation A as well as for the phase currents and voltages from thesubstation is for B (FIG. 11).

The value of the synchronization angle (δ) is calculated from thefollowing condition:I _(A) _(—) _(x) =−I _(B) _(—) _(x)   (31)where:I _(A) _(—) _(x) =I _(A) _(—) _(pre) _(—) ₁ _(e) ^(jδ) −j0.5ω₁ C _(L1) V_(A) _(—) _(pre) _(—) ₁ _(e) ^(jδ)I _(B) _(—) B _(—) _(x) =I _(B) _(—) _(pre) _(—) ₁ −j0.5ω₁ C _(L1) V_(B) _(—) _(pre) _(—) ₁

From (31) one obtains: $\begin{matrix}{{\mathbb{e}}^{j\delta} = \frac{{- {\underset{\_}{I}}_{{B\_ pre}\_ 1}} + {{j0}{.5}\omega_{1}C_{L\quad 1}{\underset{\_}{V}}_{{B\_ pre}\_ 1}}}{{\underset{\_}{I}}_{{A\_ pre}\_ 1} - {{j0}{.5}\omega_{1}C_{L\quad 1}{\underset{\_}{V}}_{{A\_ pre}\_ 1}}}} & (32)\end{matrix}$

A more precise value of the synchronization angle can be obtained byusing a long line model (with distributed parameters).

The method and a fault locator device according to any embodiment of theinvention may be used to determine distance to a fault on a section ofpower transmission line. The present invention may also be used todetermine a distance to a fault on a section of a power distributionline, or any other line or bus arranged for any of generation,transmission, distribution, control or consumption of electrical power.

FIG. 19 shows an embodiment of a device for determining the distancefrom one end, A or B, of a section of transmission line, to a fault F onthe transmission line according to the described method. The device andsystem comprises certain measuring devices such as current measuringmeans 10, 12, voltage measurement means 11, 13, measurement valueconverters, members for treatment of the calculating algorithms of themethod, indicating means for the calculated distance to fault and aprinter for printout of the calculated fault.

In the embodiment shown, measuring devices 10 and 12 for continuousmeasurement of all the phase currents, and measuring devices 11, 13 formeasurement of voltages, are arranged in both stations A and B. Themeasured values V _(A), I _(A), V _(B), I _(B) are all passed to acalculating unit 20, filtered and stored. The calculating unit isprovided with the calculating algorithms described, programmed for theprocesses needed for calculating the distance to fault. In FIG. 19 thehigh speed communication means 14 is shown arranged in respect ofreceiving communications from section end B, but could as well bearranged in respect of section end A instead. The calculating unit 20contains means (such as a means for carrying out a procedure describedin EP 506 035B1 described above) for determining whether a CT issaturated or not. The calculating unit 20 is provided with pre-faultphase currents and also with known values such as shunt capacitances andthe impedances of the line. In respect of the occurrence of a fault,information regarding the type of fault may be supplied to thecalculating unit. When the calculating unit has determined the distanceto fault, it is displayed on the device and/or sent to remotely locateddisplay means. A printout of the result may also be provided. Inaddition to signaling the fault distance, the device can producereports, in which are recorded measured values of the currents of bothlines, voltages, type of fault and other measured and/or calculatedinformation associated with a given fault at a distance. Informationabout a fault and its location may be automatically notified tooperational network centres or to automatically start calculations suchas to:

determine journey time to location;

select which repair crew shall be dispatched to site;

estimate possible time taken to execute a repair;

propose alternative arrangements for power supply;

select which vehicles or crew member may be needed;

estimate how many shifts work per crew will be required,

and the like actions.

The fault locator device and system may comprise filters for filteringthe signals, converters for sampling the signals and one or more microcomputers. The micro processor (or processors) comprises one or morecentral processing units (CPU) performing the steps of the methodaccording to the invention. This is performed with the aid of adedicated computer program, which is stored in the program memory. It isto be understood that the computer program may also be run on one ormore general purpose industrial computers or microprocessors instead ofa specially adapted computer.

The software includes computer program code elements or software codeportions that make the computer perform the method using equations,algorithms, data and calculations previously described. A part of theprogram may be stored in a processor as above, but also in a ROM, RAM,PROM or EPROM chip or similar. The program in part or in whole may alsobe stored on, or in, other suitable computer readable medium such as amagnetic disk, CD-ROM or DVD disk, hard disk, magneto-optical memorystorage means, in volatile memory, in flash memory, as firmware, orstored on a data server.

A computer program product according to an aspect of the invention maybe stored at least in part on different mediums that are computerreadable. Archive copies may be stored on standard magnetic disks, harddrives, CD or DVD disks, or magnetic tape. The databases and librariesare stored preferably on one or more local or remote data servers, butthe computer program products may, for example at different times, bestored in any of; a volatile Random Access memory (RAM) of a computer orprocessor, a hard drive, an optical or magneto-optical drive, or in atype of non-volatile memory such as a ROM, PROM, or EPROM device. Thecomputer program product may also be arranged in part as a distributedapplication capable of running on several different computers orcomputer systems at more or less the same time.

It is also noted that while the above describes exemplifying embodimentsof the invention, there are several variations and modifications whichmay be made to the disclosed solution without departing from the scopeof the present invention as defined in the appended claims.

REFERENCES

-   [1] ERIKSSON L., SAHA M. M., ROCKEFELLER G. D., An accurate fault    locator with compensation for apparent reactance in the fault    resistance resulting from remote-end infeed, IEEE Transactions on    Power Apparatus and Systems, Vol. PAS-104, No. 2, February 1985, pp.    424-436.-   [2] NOVOSEL D., HART D. G., UDREN E., GARITTY J., Unsynchronized    two-terminal fault location estimation, IEEE Transactions on Power    Delivery, Vol. 11, No. 1, January 1996, pp. 130-138.-   [3] TZIOUVARAS D. A., ROBERTS J., BENMMOUYAL G., New multi-ended    fault location design for two-or three-terminal lines, Proceedings    of Seventh International Conference on Developments in Power System    Protection, Conference Publication No.479, IEE 201, pp.395-398.-   [4] ANDERSON P. M., Power system protection, McGraw-Hill, 1999.    Appendix 1—Derivation of the Coefficients from Tables 1, 2

In classic distance relaying or in the RANZA fault locator [1] the phasequantities are used for determining the fault loop voltages. Similarly,phase currents, but compensated for the zero sequence current (in caseof single phase-to-ground faults) are used for defining the fault loopcurrents. This method is marked in TABLE 1A as the classic approach. Incontrast, in the description of the new fault location algorithm thefault loop signals (both, voltage and current) are defined in terms ofsymmetrical quantities (the symmetrical components approach—in TABLE1A). Both, the classic and symmetrical approaches are equivalent to eachother. However, the applied here symmetrical components approach isbetter since it enables to use the generalized fault loop model, whatleads to obtaining the single formula for a distance to fault, coveringdifferent fault types (appropriate coefficients, relevant for aparticular fault type are used). Moreover, the applied symmetricalcomponents approach enables to perform the compensation for shuntcapacitances individually for all the sequence quantities. TABLE IAFault loop voltage (V _(A) _(—) _(FL)) and current (I _(A) _(—) _(FL))defined by using the classic and symmetrical components approaches. Thesymmetrical components approach V _(A) _(—) _(FL) = a ₁ V _(A1) + a ₂ V_(A2) + a ₀ V _(A0) The classic approach${\underset{\_}{I}}_{A\_ FL} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A1}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A2}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{0L}}{{\underset{\_}{Z}}_{1L}}{\underset{\_}{I}}_{A0}}}$Fault type V _(A) _(—) _(FL) I _(A) _(—) _(FL) a ₁ a ₂ a ₀ a-g V _(A)_(—) _(a) I _(A) _(—) _(a) + k ₀ I _(A0) 1 1 1 b-g V _(A) _(—) _(b) I_(A) _(—) _(b) + k ₀ I _(A0) a ² a 1 c-g V _(A) _(—) _(c) I _(A) _(—)_(c) + k ₀ I _(A0) a a ² 1 a-b, a-b-g V _(A) _(—) _(a) − V _(A) _(—)_(b) I _(A) _(—) _(a) − I _(A) _(—) _(b) 1 − a ² 1 − a 0 a-b-c a-b-c-gb-c, b-c-g V _(A) _(—) _(b) − V _(A) _(—) _(c) I _(A) _(—) _(b) − I _(A)_(—) _(c) a ² − a a − a ² 0 c-a, c-a-g V _(A) _(—) _(c) − V _(A) _(—)_(a) I _(A) _(—) _(c) − I _(A) _(—) _(a) a − 1 a ² − 1 0 The signals aredefined for the fault loop seen from the substation A${\underset{\_}{k}}_{0} = {{\frac{{\underset{\_}{Z}}_{L0} - {\underset{\_}{Z}}_{L1}}{{\underset{\_}{Z}}_{L1}}\quad\underset{\_}{a}} = {\exp( {{j2\pi}/3} )}}$Examples of the Derivation of the Coefficients α ₁, α ₂, α ₀1. Single Phase-To-Ground Fault: a-g FaultI _(A) _(—) _(FL) =V _(A) _(—) _(a) =V _(A1) +V _(A2) +V _(A0)=α ₁ V_(A1)+α ₂ V _(A2)+α ₀ V _(A0)$\begin{matrix}{{\underset{\_}{I}}_{A\_ FL} = {{\underset{\_}{I}}_{A\_ a} + {{\underset{\_}{k}}_{0}{\underset{\_}{I}}_{A\quad 0}}}} \\{= {{\underset{\_}{I}}_{A\quad 1} + {{\underset{\_}{I}}_{A\quad 2}{\underset{\_}{I}}_{A\quad 0}} + {\frac{{\underset{\_}{Z}}_{L\quad 0} - {\underset{\_}{Z}}_{L\quad 1}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}}}} \\{= {{{\underset{\_}{I}}_{A\quad 1} + {\underset{\_}{I}}_{A\quad 2} + {\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}}} =}} \\{= {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}}}}\end{matrix}$

Thus: α ₁=α ₂=α ₀=12. Inter-Phase Faults: a-g, a-b-g, a-b-c, a-b-c-g Faults $\begin{matrix}{{\underset{\_}{V}}_{A\_ FL} = {{\underset{\_}{V}}_{A\_ a} - {\underset{\_}{V}}_{A\_ b}}} \\{= {( {{\underset{\_}{V}}_{A\quad 1} + {\underset{\_}{V}}_{A\quad 2} + {\underset{\_}{V}}_{A\quad 0}} ) - ( {{{\underset{\_}{a}}^{2}{\underset{\_}{V}}_{A\quad 1}} + {\underset{\_}{aV}}_{A\quad 2} + {\underset{\_}{V}}_{A\quad 0}} )}} \\{= {{{( {1 - {\underset{\_}{a}}^{2}} ){\underset{\_}{V}}_{A\quad 1}} + {( {1 - \underset{\_}{a}} ){\underset{\_}{V}}_{A\quad 2}}} =}} \\{= {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{A\quad 1}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{A\quad 2}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{A\quad 0}}}}\end{matrix}$ $\begin{matrix}{{\underset{\_}{I}}_{A\_ FL} = {{\underset{\_}{I}}_{A\_ a} - {\underset{\_}{I}}_{A\_ b}}} \\{= {( {{\underset{\_}{I}}_{A\quad 1} + {\underset{\_}{I}}_{A\quad 2} + {\underset{\_}{I}}_{A\quad 0}} ) - ( {{{\underset{\_}{a}}^{2}{\underset{\_}{I}}_{A\quad 1}} + {\underset{\_}{aI}}_{A\quad 2} + {\underset{\_}{I}}_{A\quad 0}} )}} \\{= {{{( {1 - {\underset{\_}{a}}^{2}} ){\underset{\_}{I}}_{A\quad 1}} + {( {1 - \underset{\_}{a}} ){\underset{\_}{I}}_{A\quad 2}}} =}} \\{= {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}}}}\end{matrix}$

Thus: α ₁=1−α ², α ₂=1−α, α ₁=1

TABLE 2 contains three alternative sets (Set I, Set II, Set III) of theweighting coefficients, which are used for determining a voltage dropacross a fault path. The coefficients are calculated from the boundaryconditions—relevant for a particular fault type. It is distinctive thatin all the sets the zero sequence is omitted (α _(F0)=0). It isadvantages since the zero sequence impedance of a line is considered asthe uncertain parameter. By setting α _(F0)=0 we limit adverse influenceof the uncertainty with respect to the zero sequence impedance data uponthe fault location accuracy. To be precise one has to note that thislimitation is of course partial since it is related only to determiningthe voltage drop across a fault path. In contrast, while determining thevoltage drop across a faulted line segment the zero sequence impedanceof the line is used. TABLE 2 Alternative sets of the weightingcoefficients from (5) for determining a voltage drop across the faultpath resistance Fault Set I Set II Set III type a_(F1) a_(F2) a_(F0)a_(F1) a_(F2) a_(F0) a_(F1) a_(F2) a_(F0) a-g 0 3 0 3 0 0 1.5 1.5 0 b-g0 3a 0 3a² 0 0 1.5a² 1.5a 0 c-g 0 3a² 0 3a 0 0 1.5a 1.5a² 0 a-b 0 1 − a0 1 − a² 0 0 0.5(1 − a²) 0.5(1 − a) 0 b-c 0 a − a² 0 a² − a 0 0 0.5(a² −a) 0.5(a − a²) 0 c-a 0 a² − 1 0 a − 1 0 0 0.5(a − 1) 0.5(a² − 1) 0 a-b-g1 − a² 1 − a 0 1 − a² 1 − a 0 1 − a² 1 − a 0 b-c-g a² − a a − a² 0 a² −a a − a² 0 a² − a a − a² 0 c-a-g a − 1 a² − 1 0 a − 1 a² − 1 0 a − 1 a²− 1 0 a-b-c-g 1 − a² 0 0 1 − a² 0 0 1 − a² 0 0 (a-b-c)Examples of the Derivation of the Coefficients α _(F1), α _(F2), α _(F0)FIG. 15, a-g Fault

Taking into account that in the healthy phases: I _(F) _(—) _(b)=I _(F)_(—) _(c)=0 this gives: $\begin{matrix}{{\underset{\_}{I}}_{F\quad 1} = {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {\underset{\_}{aI}}_{F\_ b} + {{\underset{\_}{a}}^{2}{\underset{\_}{I}}_{F\_ c}}} )}} \\{= {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {\underset{\_}{a}\quad 0} + {{\underset{\_}{a}}^{2}0}} )}} \\{= {\frac{1}{3}{\underset{\_}{I}}_{F\_ a}}}\end{matrix}$ $\begin{matrix}{{\underset{\_}{I}}_{F\quad 2} = {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {{\underset{\_}{a}}^{2}{\underset{\_}{I}}_{F\_ b}} + {\underset{\_}{aI}}_{F\_ c}} )}} \\{= {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {{\underset{\_}{a}}^{2}0} + {\underset{\_}{a}0}} )}} \\{= {\frac{1}{3}{\underset{\_}{I}}_{F\_ a}}}\end{matrix}$ $\begin{matrix}{{\underset{\_}{I}}_{F\quad 0} = {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {\underset{\_}{I}}_{F\_ b} + {\underset{\_}{I}}_{F\_ c}} )}} \\{= {\frac{1}{3}( {{\underset{\_}{I}}_{Fa} + 0 + 0} )}} \\{= {\frac{1}{3}{\underset{\_}{I}}_{F\_ a}}}\end{matrix}$

The sequence components are related: I _(F1)=I _(F2)=I _(F0) andfinally: I _(F)=I _(F) _(—) _(a)=3I _(F2), thus: α _(F1)=0, α _(F)=3, α_(F)=0 (as in the SET I from Table 2)

-   -   or

I _(F)=I _(F) _(—) _(a)=3I _(F1), thus: α _(F1) =3, αF2=0, α _(F)=0 (asin the SET II from Table 2)

-   -   or

I _(F)=I _(F) _(—) _(a)=1,5I _(F1)+1,5I _(F2), thus: α _(F1)=1,5, α_(F2)=1,5, α _(F)=0 (as in the SET III from Table 2)

FIG. 16 a, 16 b a-b Fault:

The fault current can be expressed as: I _(F)=I _(F) _(—) _(a) or as:${\underset{\_}{I}}_{F\quad} = {\frac{1}{2}( {{\underset{\_}{I}}_{F\_ a} - {\underset{\_}{I}}_{F\_ b}} )}$

Taking into account that in the healthy phase: IF _(—) _(c)=0 and forthe faulted phases: I _(F) _(—) _(b)=−I _(F) _(—) _(a), this gives:$\begin{matrix}{{\underset{\_}{I}}_{F\quad 1} = {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {\underset{\_}{aI}}_{F\_ b} + {{\underset{\_}{a}}^{2}{\underset{\_}{I}}_{F\_ c}}} )}} \\{= {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {\underset{\_}{a}\quad( {- {\underset{\_}{I}}_{F\_ a}} )} + {{\underset{\_}{a}}^{2}0}} )}} \\{= {\frac{1}{3}( {1 - \underset{\_}{a}} ){\underset{\_}{I}}_{F\_ a}}}\end{matrix}$ $\begin{matrix}{{\underset{\_}{I}}_{F\quad 2} = {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {{\underset{\_}{a}}^{2}{\underset{\_}{I}}_{F\_ b}} + {\underset{\_}{aI}}_{F\_ c}} )}} \\{= {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {{\underset{\_}{a}}^{2}( {\underset{\_}{I}}_{F\_ a} )} + {\underset{\_}{a}0}} )}} \\{= {\frac{1}{3}( {1 - {\underset{\_}{a}}^{2}} ){\underset{\_}{I}}_{F\_ a}}}\end{matrix}$ $\begin{matrix}{{\underset{\_}{I}}_{F\quad 0} = {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + {\underset{\_}{I}}_{F\_ b} + {\underset{\_}{I}}_{F\_ c}} )}} \\{= {\frac{1}{3}( {{\underset{\_}{I}}_{F\_ a} + ( {- {\underset{\_}{I}}_{F\_ a}} ) + 0} )}} \\{= 0}\end{matrix}$

The relation between I _(F1) and I _(F2) is thus:$\frac{{\underset{\_}{I}}_{F\quad 1}}{{\underset{\_}{I}}_{F\quad 2}} = {\frac{\frac{1}{3}( {1 - \underset{\_}{a}} ){\underset{\_}{I}}_{F\_ a}}{\frac{1}{3}( {1 - {\underset{\_}{a}}^{2}} ){\underset{\_}{I}}_{F\_ a}} = \frac{( {1 - \underset{\_}{a}} )}{( {1 - {\underset{\_}{a}}^{2}} )}}$

Finally:${\underset{\_}{I}}_{F} = {{\underset{\_}{I}}_{F\_ a} = {{\frac{3}{( {1 - {\underset{\_}{a}}^{2}} )}{\underset{\_}{I}}_{F\quad 2}} = {( {1 - \underset{\_}{a}} ){\underset{\_}{I}}_{F\quad 2}}}}$

thus: α _(F1)=0, α _(F2)−1−α, α _(F0)=0 (as in the SET I from Table 2)or${\underset{\_}{I}}_{F} = {{\underset{\_}{I}}_{F\_ a} = {{\frac{3}{( {1 - \underset{\_}{a}} )}{\underset{\_}{I}}_{F\quad 1}} = {( {1 - {\underset{\_}{a}}^{2}} ){\underset{\_}{I}}_{F\quad 1}}}}$

thus: α _(F1)=1−α ², α _(F2)=0, α _(F0)=0 (as in the SET II from Table2) or${{\underset{\_}{I}}_{F} = 0},{{5{\underset{\_}{I}}_{F\_ a}} + 0},{{5{\underset{\_}{I}}_{F\_ a}} = {{{\frac{1,5}{( {1 - {\underset{\_}{a}}^{2}} )}{\underset{\_}{I}}_{F\quad 2}} + {\frac{1,5}{( {1 - \underset{\_}{a}} )}{\underset{\_}{I}}_{F\quad 1}}} = 0}},{{5( {1 - \underset{\_}{a}} ){\underset{\_}{I}}_{F\quad 2}} + 0},{5( {1 - {\underset{\_}{a}}^{2}} ){\underset{\_}{I}}_{F\quad 1}}$

thus: α _(F1)=0,5(1−α ²), α _(F2)=0,5(1−α), α _(F0)=0 (as in the SET IIIfrom Table 2)See FIG. 17, (a-b-g) Fault: $\begin{matrix}{{\underset{\_}{I}}_{F} = {{\underset{\_}{I}}_{F\_ a} - {\underset{\_}{I}}_{F\_ b}}} \\{= {{( {{\underset{\_}{I}}_{F\quad 1} + {\underset{\_}{I}}_{F\quad 2} + {\underset{\_}{I}}_{F\quad 0}} ) - ( {{{\underset{\_}{a}}^{2}{\underset{\_}{I}}_{F\quad 1}} + {\underset{\_}{aI}}_{F\quad 2} + {\underset{\_}{I}}_{F\quad 0}} )} =}} \\{= {{( {1 - {\underset{\_}{a}}^{2}} ){\underset{\_}{I}}_{F\quad 1}} + {( {1 - \underset{\_}{a}} ){\underset{\_}{I}}_{F\quad 2}}}}\end{matrix}$

Thus: α _(F1)=1−α ², α _(F2)=1−α, α _(F0)=0 (as in the SETs I, II, IIIfrom Table 2)

See FIG. 18 a, 18 b, (a-b-c) or (a-b-c-g) Symmetrical Faults:

Taking the first two phases (a, b) for composing the voltage drop acrossa fault path one obtains: $\begin{matrix}{{\underset{\_}{I}}_{F} = {{\underset{\_}{I}}_{F\_ a} - {\underset{\_}{I}}_{F\_ b}}} \\{= {{( {{\underset{\_}{I}}_{F\quad 1} + {\underset{\_}{I}}_{F\quad 2} + {\underset{\_}{I}}_{F\quad 0}} ) - ( {{{\underset{\_}{a}}^{2}{\underset{\_}{I}}_{F\quad 1}} + {\underset{\_}{aI}}_{F\quad 2} + {\underset{\_}{I}}_{F\quad 0}} )} =}} \\{= {{( {1 - {\underset{\_}{a}}^{2}} ){\underset{\_}{I}}_{F\quad 1}} + {( {1 - \underset{\_}{a}} ){\underset{\_}{I}}_{F\quad 2}}}}\end{matrix}$

Thus:α _(F1)=1−α ², α _(F2)=1−α, α _(F0=0)

Additionally, if a fault is ideally symmetrical the positive sequence isthe only component, which is present in the signals. Therefore, we have:

α _(F1)=1−α ², α _(F2)=0, α _(F0)=0 (as in the SETs I, II, III fromTable 2).

1. A method to locate a fault in a section of a transmission line usingmeasurements of current, voltage and angles between the phases at afirst and a second end of said section, the method comprising after theoccurrence of a fault along the section, calculating a distance to afault dependent on a fault current measured at one of said first andsecond ends and phase voltages measured at both of said first and secondends where the distance to fault is calculated from the end where thefault current is measured.
 2. The method according to claim 1, whereinthe distance to a fault is calculated by means of a formula such as:$d_{A} = \frac{\begin{matrix}{{{{real}( {\underset{\_}{A}}_{v} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {\underset{\_}{A}}_{v} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}\end{matrix}}{\begin{matrix}{{{{real}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}\end{matrix}}$ where:${{\underset{\_}{A}}_{v}\quad{is}\quad\ldots\quad{and}\quad{\underset{\_}{A}}_{v}} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{A\quad 0}{\mathbb{e}}^{j\delta}}}$${{\underset{\_}{A}}_{i}\quad{is}\quad\ldots\quad{and}\quad{\underset{\_}{A}}_{i}} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}{\mathbb{e}}^{j\delta}}}$${{\underset{\_}{M}}_{1\quad A}\quad{is}\quad\ldots\quad{and}\quad{\underset{\_}{M}}_{1\quad A}} = {\frac{{{- {\underset{\_}{V}}_{A\quad 1}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{B\quad 1}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{A\quad 1}{\mathbb{e}}^{j\delta}}}$${{\underset{\_}{M}}_{2\quad A}\quad{is}\quad\ldots\quad{and}\quad M_{2\quad A}} = {\frac{{{- {\underset{\_}{V}}_{A\quad 2}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{B\quad 2}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}}}$Z _(L1)—impedance of a whole line for the positive (negative) sequence,l—total line length, α ₁, α ₂, α ₀, α _(F1), αF2—are coefficientsdependent on a fault type, wherein d_(B) is calculated correspondinglyby substituting the values measured at end A with values measured at endB and vice versa.
 3. The method according to claim 1, further comprisingcalculating in the case of a non-zero synchronisation angle δ=0 a valuefor a term e^(jδ) using a formula such as:${\mathbb{e}}^{j\delta} = \frac{{- {\underset{\_}{I}}_{{B\_ pre}\_ 1}} + {{j0}{.5}\omega_{1}C_{L\quad 1}{\underset{\_}{V}}_{{B\_ pre}\_ 1}}}{{\underset{\_}{I}}_{{A\_ pre}\_ 1} - {{j0}{.5}\omega_{1}C_{L\quad 1}{\underset{\_}{V}}_{{A\_ pre}\_ 1}}}$where:I _(A) _(—) _(x) =I _(A) _(—) _(pre) _(—) ₁ _(e) ^(jδ) −j0.5ω₁ C _(L1) V_(A) _(—) _(pre) _(—) ₁ _(e) ^(jδ)I _(B) _(—) _(x) =I _(B) _(—) _(pre) _(—) ₁ −j0.5ω₁ C _(L1) V _(B) _(—)_(pre) _(—) ₁ I _(A) _(—) _(pre) _(—) _(a), I _(A) _(—) _(pre) _(—)_(b), I _(A) _(—) _(pre) _(—) _(c) are the pre-fault currents fromphases (a, b, c) at the station A, I _(A) _(—) _(pre) _(—) ₁ is thepositive sequence phasor, wherein, when calculating at end B, index A issubstituted with index B and vice versa.
 4. The method according toclaim 1, further comprising calculating a compensation value for a shuntcapacitance of said section of a line according to a formula such as:$d_{A} = \frac{\begin{matrix}{{{{real}( {\underset{\_}{A}}_{v} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {\underset{\_}{A}}_{v} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}\end{matrix}}{\begin{matrix}{{{{real}( {{\underset{\_}{Z}}_{L\quad 1}^{long}{\underset{\_}{A}}_{i\_ comp}} )}\quad{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {{\underset{\_}{Z}}_{L\quad 1}^{long}{\underset{\_}{A}}_{i\_ comp}} )}\quad{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}}\end{matrix}}$ where:${{\underset{\_}{A}}_{i\_ comp} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1{\_ comp}}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2{\_ comp}}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}^{long}}{{\underset{\_}{Z}}_{L\quad 1}^{long}}{\underset{\_}{I}}_{A\quad 0{\_ comp}}{\mathbb{e}}^{j\delta}}}},$I _(A1) _(—) _(comp), I _(A2) _(—) _(comp), I _(A0) _(—) _(comp)—are thepositive, negative and zero sequence currents after the compensation,${\underset{\_}{Z}\frac{long}{L\quad 0}} -$ is the positive sequenceimpedance of a line with taking into account the distributed long linemodel $( {{\underset{\_}{Z}\frac{long}{L\quad 0}} -} $ asabove, but for the zero sequence, wherein d_(B-comp) is calculatedcorrespondingly by substituting the values measured at end A with valuesmeasured at end B and vice versa.
 5. The method according to claim 1,further comprising: detecting that a current transformer at one of saidfirst and second ends on said line is saturated, determining which ofthe current transformers is saturated, calculating a distance to a faultdependent on a fault current measured at the end containing thenon-saturated current transformer and phase voltages measured at both ofsaid first and second ends.
 6. A device for fault location in a sectionof a transmission line using measurements of current, voltage and anglesbetween the phases at a first and a second end of said section,including calculating means and means for storing said measurements ofcurrent, voltage, and angles between the phases, means for calculating adistance to a fault dependent on a fault current measured at one of saidfirst and second ends and phase voltages measured at both of said firstand second ends where the distance to fault is calculated from the endwhere the fault current is measured.
 7. The device according to claim 6,wherein the means for calculating the distance to a fault includes aformula such as: $d_{A} = \frac{\begin{matrix}{{{{real}( {\underset{\_}{A}}_{v} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2A}}} )}} -} \\{{{imag}( {\underset{\_}{A}}_{v} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2A}}} )}}\end{matrix}}{\begin{matrix}{{{{real}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1\quad A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2\quad A}}} )}} -} \\{{{imag}( {{\underset{\_}{Z}}_{L\quad 1}{\underset{\_}{A}}_{i}} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2A}}} )}}\end{matrix}}$ where:${{\underset{\_}{A}}_{v}\quad{is}\quad\ldots\quad{and}\quad{\underset{\_}{A}}_{v}} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{A\quad 0}{\mathbb{e}}^{j\delta}}}$${{\underset{\_}{A}}_{i}\quad{is}\quad\ldots\quad{and}\quad{\underset{\_}{A}}_{i}} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}}{{\underset{\_}{Z}}_{L\quad 1}}{\underset{\_}{I}}_{A\quad 0}{\mathbb{e}}^{j\delta}}}$${{\underset{\_}{M}}_{1A}\quad{is}\quad\ldots\quad{and}\quad{\underset{\_}{M}}_{1A}} = {\frac{{{- {\underset{\_}{V}}_{A\quad 1}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{B\quad 1}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{A1}{\mathbb{e}}^{j\delta}}}$${{\underset{\_}{M}}_{2A}\quad{is}\quad\ldots\quad{and}\quad{\underset{\_}{M}}_{2A}} = {\frac{{{- {\underset{\_}{V}}_{A\quad 2}}{\mathbb{e}}^{j\delta}} + {\underset{\_}{V}}_{B\quad 2}}{{\underset{\_}{Z}}_{L\quad 1}} + {{\underset{\_}{I}}_{A\quad 2}{\mathbb{e}}^{j\delta}}}$Z _(L1)—impedance of a whole line for the positive (negative) sequence,l—total line length, α ₁, α ₂, α ₀, α _(F1), α _(F2)—are coefficientsdependent on a fault type, wherein d_(B) is calculated correspondinglyby substituting the values measured at end A with values measured at endB and vice versa.
 8. The device according to claim 6, wherein the meansfor calculating the distance to a fault includes a formula to calculatea compensation value for a shunt capacitance of said section of a linesuch as: $d_{A\_ comp} = \frac{\begin{matrix}{{{{real}( {\underset{\_}{A}}_{v} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2A}}} )}} -} \\{{{imag}( {\underset{\_}{A}}_{v} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2A}}} )}}\end{matrix}}{\begin{matrix}{{{{real}( {{\underset{\_}{Z}}_{L\quad 1}^{long}{\underset{\_}{A}}_{i\_ comp}} )}{{imag}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2A}}} )}} -} \\{{{imag}( {{\underset{\_}{Z}}_{L\quad 1}^{long}{\underset{\_}{A}}_{i\_ comp}} )}{{real}( {{{\underset{\_}{a}}_{F\quad 1}{\underset{\_}{M}}_{1A}} + {{\underset{\_}{a}}_{F\quad 2}{\underset{\_}{M}}_{2A}}} )}}\end{matrix}}$ where:${{\underset{\_}{A}}_{i\_ comp} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\quad 1{\_ comp}}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\quad 2{\_ comp}}{\mathbb{e}}^{j\delta}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{L\quad 0}^{long}}{{\underset{\_}{Z}}_{L\quad 1}^{long}}{\underset{\_}{I}}_{A\quad 0{\_ comp}}{\mathbb{e}}^{j\delta}}}},$I _(A1) _(—) _(comp), I _(A2) _(—) _(comp), I _(A0) _(—) _(comp)—are thepositive, negative and zero sequence currents after the compensation,${\underset{\_}{Z}\frac{long}{L\quad 0}} -$ is the positive sequenceimpedance of a line with taking into account the distributed long linemodel $( {{{- \underset{\_}{Z}}\frac{long}{L\quad 0}} -} $as above, but for the zero sequence, wherein is calculatedcorrespondingly by substituting the values measured at end A with valuesmeasured at end B and vice versa.
 9. The device according to claim 6,further comprising: detecting that a current transformer at one of saidfirst and second ends on said line is saturated, means for determiningwhich of the current transformers is saturated, and means forcalculating a distance to a fault dependent on a fault current measuredat the end containing the non-saturated current transformer and phasevoltages measured at both of said first and second ends.
 10. A computerprogram product comprising computer code means and/or software codeportions for making a computer or processor perform the steps accordingto claim
 1. 11. The computer program product according to claim 10recorded on one or more computer readable media.
 12. Use of a faultlocator device according to claim 6 to calculate a distance to a faulton a section of a line in an electrical power transmission anddistribution system.
 13. Use of a fault locator device according toclaim 6 to provide information to carry out repair and/or maintenance ofa section of a line in an electrical power transmission and distributionsystem.]